Verification of a Conditional expectation

Question

I need to check if the following formula is correct.

$$ \mathrm{E}(Z \vert X=x)=\sum_{y} \mathrm{E}(Z \vert X=x,Y=y)\mathrm{P}(Y=y \vert X=x) $$

Just as a reference, Law of total expectation states that $\mathrm{E}(Z)= \sum_{x} \mathrm{E}[Z \vert X=x]\mathrm{P}(X=x)$

Any help would be much appreciated :)

Welcome to MSE! It helps readability to format using MathJax (see FAQ). Also, can you share what you have tried? Did you provide enough information? Regards — Amzoti, Aug 26 '13 at 04:25

score 0 · Accepted Answer · answered Aug 28 '13 at 02:10

It appears that all that was done was to condition $(Z|X)$ itself on $Y$. So as long as the expectation of the twice-conditioned interior is taken against all $Y|X=x$ then that outermost conditioning should drop and we are back where we started. This is sometimes called the law of iterated expectations.

Verification of a Conditional expectation

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